Properties

Label 8640.2.a.df.1.1
Level $8640$
Weight $2$
Character 8640.1
Self dual yes
Analytic conductor $68.991$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8640,2,Mod(1,8640)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8640, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8640.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8640 = 2^{6} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8640.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(68.9907473464\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 135)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.30278\) of defining polynomial
Character \(\chi\) \(=\) 8640.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} -2.60555 q^{7} +O(q^{10})\) \(q+1.00000 q^{5} -2.60555 q^{7} +4.60555 q^{11} -6.60555 q^{13} -1.60555 q^{17} -3.60555 q^{19} +3.00000 q^{23} +1.00000 q^{25} +1.39445 q^{29} -5.60555 q^{31} -2.60555 q^{35} -2.00000 q^{37} +4.60555 q^{41} -0.605551 q^{43} +9.21110 q^{47} -0.211103 q^{49} +1.60555 q^{53} +4.60555 q^{55} -1.39445 q^{59} +4.21110 q^{61} -6.60555 q^{65} +0.788897 q^{67} -7.39445 q^{71} +12.6056 q^{73} -12.0000 q^{77} -11.6056 q^{79} +3.00000 q^{83} -1.60555 q^{85} +13.8167 q^{89} +17.2111 q^{91} -3.60555 q^{95} +8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} + 2 q^{7} + 2 q^{11} - 6 q^{13} + 4 q^{17} + 6 q^{23} + 2 q^{25} + 10 q^{29} - 4 q^{31} + 2 q^{35} - 4 q^{37} + 2 q^{41} + 6 q^{43} + 4 q^{47} + 14 q^{49} - 4 q^{53} + 2 q^{55} - 10 q^{59} - 6 q^{61} - 6 q^{65} + 16 q^{67} - 22 q^{71} + 18 q^{73} - 24 q^{77} - 16 q^{79} + 6 q^{83} + 4 q^{85} + 6 q^{89} + 20 q^{91} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −2.60555 −0.984806 −0.492403 0.870367i \(-0.663881\pi\)
−0.492403 + 0.870367i \(0.663881\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.60555 1.38863 0.694313 0.719673i \(-0.255708\pi\)
0.694313 + 0.719673i \(0.255708\pi\)
\(12\) 0 0
\(13\) −6.60555 −1.83205 −0.916025 0.401121i \(-0.868621\pi\)
−0.916025 + 0.401121i \(0.868621\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.60555 −0.389403 −0.194702 0.980863i \(-0.562374\pi\)
−0.194702 + 0.980863i \(0.562374\pi\)
\(18\) 0 0
\(19\) −3.60555 −0.827170 −0.413585 0.910465i \(-0.635724\pi\)
−0.413585 + 0.910465i \(0.635724\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.00000 0.625543 0.312772 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.39445 0.258943 0.129471 0.991583i \(-0.458672\pi\)
0.129471 + 0.991583i \(0.458672\pi\)
\(30\) 0 0
\(31\) −5.60555 −1.00679 −0.503393 0.864057i \(-0.667915\pi\)
−0.503393 + 0.864057i \(0.667915\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.60555 −0.440419
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.60555 0.719266 0.359633 0.933094i \(-0.382902\pi\)
0.359633 + 0.933094i \(0.382902\pi\)
\(42\) 0 0
\(43\) −0.605551 −0.0923457 −0.0461729 0.998933i \(-0.514703\pi\)
−0.0461729 + 0.998933i \(0.514703\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.21110 1.34358 0.671789 0.740743i \(-0.265526\pi\)
0.671789 + 0.740743i \(0.265526\pi\)
\(48\) 0 0
\(49\) −0.211103 −0.0301575
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.60555 0.220539 0.110270 0.993902i \(-0.464829\pi\)
0.110270 + 0.993902i \(0.464829\pi\)
\(54\) 0 0
\(55\) 4.60555 0.621012
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.39445 −0.181542 −0.0907709 0.995872i \(-0.528933\pi\)
−0.0907709 + 0.995872i \(0.528933\pi\)
\(60\) 0 0
\(61\) 4.21110 0.539176 0.269588 0.962976i \(-0.413112\pi\)
0.269588 + 0.962976i \(0.413112\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.60555 −0.819318
\(66\) 0 0
\(67\) 0.788897 0.0963792 0.0481896 0.998838i \(-0.484655\pi\)
0.0481896 + 0.998838i \(0.484655\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −7.39445 −0.877560 −0.438780 0.898595i \(-0.644589\pi\)
−0.438780 + 0.898595i \(0.644589\pi\)
\(72\) 0 0
\(73\) 12.6056 1.47537 0.737684 0.675146i \(-0.235919\pi\)
0.737684 + 0.675146i \(0.235919\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −12.0000 −1.36753
\(78\) 0 0
\(79\) −11.6056 −1.30573 −0.652863 0.757476i \(-0.726432\pi\)
−0.652863 + 0.757476i \(0.726432\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.00000 0.329293 0.164646 0.986353i \(-0.447352\pi\)
0.164646 + 0.986353i \(0.447352\pi\)
\(84\) 0 0
\(85\) −1.60555 −0.174146
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 13.8167 1.46456 0.732281 0.681002i \(-0.238456\pi\)
0.732281 + 0.681002i \(0.238456\pi\)
\(90\) 0 0
\(91\) 17.2111 1.80421
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.60555 −0.369922
\(96\) 0 0
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 7.00000 0.670478 0.335239 0.942133i \(-0.391183\pi\)
0.335239 + 0.942133i \(0.391183\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 15.2111 1.43094 0.715470 0.698643i \(-0.246213\pi\)
0.715470 + 0.698643i \(0.246213\pi\)
\(114\) 0 0
\(115\) 3.00000 0.279751
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 4.18335 0.383487
\(120\) 0 0
\(121\) 10.2111 0.928282
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −19.2111 −1.70471 −0.852355 0.522964i \(-0.824826\pi\)
−0.852355 + 0.522964i \(0.824826\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) 9.39445 0.814602
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −16.8167 −1.43674 −0.718372 0.695659i \(-0.755112\pi\)
−0.718372 + 0.695659i \(0.755112\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −30.4222 −2.54403
\(144\) 0 0
\(145\) 1.39445 0.115803
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 23.0278 1.88651 0.943254 0.332073i \(-0.107748\pi\)
0.943254 + 0.332073i \(0.107748\pi\)
\(150\) 0 0
\(151\) 14.4222 1.17366 0.586831 0.809709i \(-0.300375\pi\)
0.586831 + 0.809709i \(0.300375\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −5.60555 −0.450249
\(156\) 0 0
\(157\) 17.8167 1.42192 0.710962 0.703231i \(-0.248260\pi\)
0.710962 + 0.703231i \(0.248260\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −7.81665 −0.616039
\(162\) 0 0
\(163\) −2.00000 −0.156652 −0.0783260 0.996928i \(-0.524958\pi\)
−0.0783260 + 0.996928i \(0.524958\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3.00000 0.232147 0.116073 0.993241i \(-0.462969\pi\)
0.116073 + 0.993241i \(0.462969\pi\)
\(168\) 0 0
\(169\) 30.6333 2.35641
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −10.8167 −0.822375 −0.411187 0.911551i \(-0.634886\pi\)
−0.411187 + 0.911551i \(0.634886\pi\)
\(174\) 0 0
\(175\) −2.60555 −0.196961
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −21.2111 −1.58539 −0.792696 0.609617i \(-0.791323\pi\)
−0.792696 + 0.609617i \(0.791323\pi\)
\(180\) 0 0
\(181\) 7.00000 0.520306 0.260153 0.965567i \(-0.416227\pi\)
0.260153 + 0.965567i \(0.416227\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) −7.39445 −0.540736
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 12.4222 0.898839 0.449420 0.893321i \(-0.351631\pi\)
0.449420 + 0.893321i \(0.351631\pi\)
\(192\) 0 0
\(193\) 0.183346 0.0131975 0.00659877 0.999978i \(-0.497900\pi\)
0.00659877 + 0.999978i \(0.497900\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −22.8167 −1.62562 −0.812810 0.582529i \(-0.802063\pi\)
−0.812810 + 0.582529i \(0.802063\pi\)
\(198\) 0 0
\(199\) −1.21110 −0.0858528 −0.0429264 0.999078i \(-0.513668\pi\)
−0.0429264 + 0.999078i \(0.513668\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −3.63331 −0.255008
\(204\) 0 0
\(205\) 4.60555 0.321666
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −16.6056 −1.14863
\(210\) 0 0
\(211\) 8.81665 0.606963 0.303482 0.952837i \(-0.401851\pi\)
0.303482 + 0.952837i \(0.401851\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −0.605551 −0.0412983
\(216\) 0 0
\(217\) 14.6056 0.991489
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 10.6056 0.713407
\(222\) 0 0
\(223\) −10.0000 −0.669650 −0.334825 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −11.7889 −0.782457 −0.391228 0.920294i \(-0.627950\pi\)
−0.391228 + 0.920294i \(0.627950\pi\)
\(228\) 0 0
\(229\) −8.21110 −0.542605 −0.271302 0.962494i \(-0.587454\pi\)
−0.271302 + 0.962494i \(0.587454\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) 9.21110 0.600866
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 15.2111 0.983924 0.491962 0.870617i \(-0.336280\pi\)
0.491962 + 0.870617i \(0.336280\pi\)
\(240\) 0 0
\(241\) 13.7889 0.888221 0.444110 0.895972i \(-0.353520\pi\)
0.444110 + 0.895972i \(0.353520\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −0.211103 −0.0134868
\(246\) 0 0
\(247\) 23.8167 1.51542
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 27.6333 1.74420 0.872099 0.489329i \(-0.162758\pi\)
0.872099 + 0.489329i \(0.162758\pi\)
\(252\) 0 0
\(253\) 13.8167 0.868646
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1.18335 −0.0738151 −0.0369076 0.999319i \(-0.511751\pi\)
−0.0369076 + 0.999319i \(0.511751\pi\)
\(258\) 0 0
\(259\) 5.21110 0.323802
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.78890 0.171971 0.0859854 0.996296i \(-0.472596\pi\)
0.0859854 + 0.996296i \(0.472596\pi\)
\(264\) 0 0
\(265\) 1.60555 0.0986282
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −3.21110 −0.195784 −0.0978922 0.995197i \(-0.531210\pi\)
−0.0978922 + 0.995197i \(0.531210\pi\)
\(270\) 0 0
\(271\) 31.2389 1.89763 0.948813 0.315839i \(-0.102286\pi\)
0.948813 + 0.315839i \(0.102286\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 4.60555 0.277725
\(276\) 0 0
\(277\) −7.02776 −0.422257 −0.211128 0.977458i \(-0.567714\pi\)
−0.211128 + 0.977458i \(0.567714\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 19.8167 1.18216 0.591081 0.806612i \(-0.298701\pi\)
0.591081 + 0.806612i \(0.298701\pi\)
\(282\) 0 0
\(283\) −3.39445 −0.201779 −0.100890 0.994898i \(-0.532169\pi\)
−0.100890 + 0.994898i \(0.532169\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) 0 0
\(289\) −14.4222 −0.848365
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −7.18335 −0.419656 −0.209828 0.977738i \(-0.567290\pi\)
−0.209828 + 0.977738i \(0.567290\pi\)
\(294\) 0 0
\(295\) −1.39445 −0.0811879
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −19.8167 −1.14603
\(300\) 0 0
\(301\) 1.57779 0.0909426
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 4.21110 0.241127
\(306\) 0 0
\(307\) −8.42221 −0.480681 −0.240340 0.970689i \(-0.577259\pi\)
−0.240340 + 0.970689i \(0.577259\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 7.81665 0.443242 0.221621 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(312\) 0 0
\(313\) −19.6333 −1.10974 −0.554870 0.831937i \(-0.687232\pi\)
−0.554870 + 0.831937i \(0.687232\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −7.60555 −0.427170 −0.213585 0.976924i \(-0.568514\pi\)
−0.213585 + 0.976924i \(0.568514\pi\)
\(318\) 0 0
\(319\) 6.42221 0.359574
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 5.78890 0.322103
\(324\) 0 0
\(325\) −6.60555 −0.366410
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −24.0000 −1.32316
\(330\) 0 0
\(331\) −29.2111 −1.60559 −0.802794 0.596257i \(-0.796654\pi\)
−0.802794 + 0.596257i \(0.796654\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0.788897 0.0431021
\(336\) 0 0
\(337\) 6.60555 0.359827 0.179914 0.983682i \(-0.442418\pi\)
0.179914 + 0.983682i \(0.442418\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −25.8167 −1.39805
\(342\) 0 0
\(343\) 18.7889 1.01451
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 30.4222 1.63315 0.816575 0.577240i \(-0.195870\pi\)
0.816575 + 0.577240i \(0.195870\pi\)
\(348\) 0 0
\(349\) 31.8444 1.70459 0.852296 0.523060i \(-0.175209\pi\)
0.852296 + 0.523060i \(0.175209\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −21.6333 −1.15142 −0.575712 0.817652i \(-0.695275\pi\)
−0.575712 + 0.817652i \(0.695275\pi\)
\(354\) 0 0
\(355\) −7.39445 −0.392457
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 9.63331 0.508427 0.254213 0.967148i \(-0.418183\pi\)
0.254213 + 0.967148i \(0.418183\pi\)
\(360\) 0 0
\(361\) −6.00000 −0.315789
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 12.6056 0.659805
\(366\) 0 0
\(367\) −2.60555 −0.136009 −0.0680043 0.997685i \(-0.521663\pi\)
−0.0680043 + 0.997685i \(0.521663\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −4.18335 −0.217189
\(372\) 0 0
\(373\) 25.2111 1.30538 0.652691 0.757624i \(-0.273640\pi\)
0.652691 + 0.757624i \(0.273640\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −9.21110 −0.474396
\(378\) 0 0
\(379\) −21.6056 −1.10980 −0.554901 0.831916i \(-0.687244\pi\)
−0.554901 + 0.831916i \(0.687244\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −24.6333 −1.25870 −0.629352 0.777121i \(-0.716679\pi\)
−0.629352 + 0.777121i \(0.716679\pi\)
\(384\) 0 0
\(385\) −12.0000 −0.611577
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 25.8167 1.30896 0.654478 0.756081i \(-0.272888\pi\)
0.654478 + 0.756081i \(0.272888\pi\)
\(390\) 0 0
\(391\) −4.81665 −0.243589
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −11.6056 −0.583939
\(396\) 0 0
\(397\) 5.39445 0.270740 0.135370 0.990795i \(-0.456778\pi\)
0.135370 + 0.990795i \(0.456778\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) 37.0278 1.84448
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −9.21110 −0.456577
\(408\) 0 0
\(409\) 5.00000 0.247234 0.123617 0.992330i \(-0.460551\pi\)
0.123617 + 0.992330i \(0.460551\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 3.63331 0.178783
\(414\) 0 0
\(415\) 3.00000 0.147264
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 23.0278 1.12498 0.562490 0.826804i \(-0.309844\pi\)
0.562490 + 0.826804i \(0.309844\pi\)
\(420\) 0 0
\(421\) −5.42221 −0.264262 −0.132131 0.991232i \(-0.542182\pi\)
−0.132131 + 0.991232i \(0.542182\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1.60555 −0.0778807
\(426\) 0 0
\(427\) −10.9722 −0.530984
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −4.18335 −0.201505 −0.100752 0.994912i \(-0.532125\pi\)
−0.100752 + 0.994912i \(0.532125\pi\)
\(432\) 0 0
\(433\) 22.2389 1.06873 0.534366 0.845253i \(-0.320551\pi\)
0.534366 + 0.845253i \(0.320551\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −10.8167 −0.517431
\(438\) 0 0
\(439\) 27.6056 1.31754 0.658771 0.752344i \(-0.271077\pi\)
0.658771 + 0.752344i \(0.271077\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −24.6333 −1.17036 −0.585182 0.810902i \(-0.698977\pi\)
−0.585182 + 0.810902i \(0.698977\pi\)
\(444\) 0 0
\(445\) 13.8167 0.654972
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −38.2389 −1.80460 −0.902302 0.431105i \(-0.858124\pi\)
−0.902302 + 0.431105i \(0.858124\pi\)
\(450\) 0 0
\(451\) 21.2111 0.998792
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 17.2111 0.806869
\(456\) 0 0
\(457\) −13.2111 −0.617989 −0.308995 0.951064i \(-0.599993\pi\)
−0.308995 + 0.951064i \(0.599993\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 21.6333 1.00756 0.503782 0.863831i \(-0.331942\pi\)
0.503782 + 0.863831i \(0.331942\pi\)
\(462\) 0 0
\(463\) −0.788897 −0.0366632 −0.0183316 0.999832i \(-0.505835\pi\)
−0.0183316 + 0.999832i \(0.505835\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.2111 0.565062 0.282531 0.959258i \(-0.408826\pi\)
0.282531 + 0.959258i \(0.408826\pi\)
\(468\) 0 0
\(469\) −2.05551 −0.0949148
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −2.78890 −0.128234
\(474\) 0 0
\(475\) −3.60555 −0.165434
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 37.8167 1.72789 0.863944 0.503589i \(-0.167987\pi\)
0.863944 + 0.503589i \(0.167987\pi\)
\(480\) 0 0
\(481\) 13.2111 0.602374
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 8.00000 0.363261
\(486\) 0 0
\(487\) −29.8167 −1.35112 −0.675561 0.737304i \(-0.736098\pi\)
−0.675561 + 0.737304i \(0.736098\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 27.2111 1.22802 0.614010 0.789298i \(-0.289555\pi\)
0.614010 + 0.789298i \(0.289555\pi\)
\(492\) 0 0
\(493\) −2.23886 −0.100833
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 19.2666 0.864226
\(498\) 0 0
\(499\) 20.3944 0.912981 0.456490 0.889728i \(-0.349106\pi\)
0.456490 + 0.889728i \(0.349106\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −2.57779 −0.114938 −0.0574691 0.998347i \(-0.518303\pi\)
−0.0574691 + 0.998347i \(0.518303\pi\)
\(504\) 0 0
\(505\) 12.0000 0.533993
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −24.8444 −1.10121 −0.550605 0.834766i \(-0.685603\pi\)
−0.550605 + 0.834766i \(0.685603\pi\)
\(510\) 0 0
\(511\) −32.8444 −1.45295
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) 42.4222 1.86573
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 28.6056 1.25323 0.626616 0.779328i \(-0.284439\pi\)
0.626616 + 0.779328i \(0.284439\pi\)
\(522\) 0 0
\(523\) −30.6056 −1.33829 −0.669144 0.743133i \(-0.733339\pi\)
−0.669144 + 0.743133i \(0.733339\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 9.00000 0.392046
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −30.4222 −1.31773
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −0.972244 −0.0418775
\(540\) 0 0
\(541\) 40.4222 1.73789 0.868943 0.494912i \(-0.164800\pi\)
0.868943 + 0.494912i \(0.164800\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 7.00000 0.299847
\(546\) 0 0
\(547\) −0.605551 −0.0258915 −0.0129458 0.999916i \(-0.504121\pi\)
−0.0129458 + 0.999916i \(0.504121\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −5.02776 −0.214190
\(552\) 0 0
\(553\) 30.2389 1.28589
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 9.63331 0.408176 0.204088 0.978953i \(-0.434577\pi\)
0.204088 + 0.978953i \(0.434577\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 0 0
\(565\) 15.2111 0.639936
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −16.1833 −0.678441 −0.339221 0.940707i \(-0.610163\pi\)
−0.339221 + 0.940707i \(0.610163\pi\)
\(570\) 0 0
\(571\) −28.4500 −1.19059 −0.595297 0.803506i \(-0.702966\pi\)
−0.595297 + 0.803506i \(0.702966\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 3.00000 0.125109
\(576\) 0 0
\(577\) 6.18335 0.257416 0.128708 0.991683i \(-0.458917\pi\)
0.128708 + 0.991683i \(0.458917\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −7.81665 −0.324289
\(582\) 0 0
\(583\) 7.39445 0.306247
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 21.0000 0.866763 0.433381 0.901211i \(-0.357320\pi\)
0.433381 + 0.901211i \(0.357320\pi\)
\(588\) 0 0
\(589\) 20.2111 0.832784
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 29.2389 1.20070 0.600348 0.799739i \(-0.295029\pi\)
0.600348 + 0.799739i \(0.295029\pi\)
\(594\) 0 0
\(595\) 4.18335 0.171500
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 22.6056 0.923638 0.461819 0.886974i \(-0.347197\pi\)
0.461819 + 0.886974i \(0.347197\pi\)
\(600\) 0 0
\(601\) −10.6333 −0.433742 −0.216871 0.976200i \(-0.569585\pi\)
−0.216871 + 0.976200i \(0.569585\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 10.2111 0.415140
\(606\) 0 0
\(607\) 24.6056 0.998709 0.499354 0.866398i \(-0.333571\pi\)
0.499354 + 0.866398i \(0.333571\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −60.8444 −2.46150
\(612\) 0 0
\(613\) 28.8444 1.16501 0.582507 0.812825i \(-0.302072\pi\)
0.582507 + 0.812825i \(0.302072\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −38.4500 −1.54794 −0.773969 0.633224i \(-0.781731\pi\)
−0.773969 + 0.633224i \(0.781731\pi\)
\(618\) 0 0
\(619\) −35.6333 −1.43222 −0.716112 0.697986i \(-0.754080\pi\)
−0.716112 + 0.697986i \(0.754080\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −36.0000 −1.44231
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 3.21110 0.128035
\(630\) 0 0
\(631\) −6.02776 −0.239961 −0.119981 0.992776i \(-0.538283\pi\)
−0.119981 + 0.992776i \(0.538283\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −19.2111 −0.762369
\(636\) 0 0
\(637\) 1.39445 0.0552501
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 24.0000 0.947943 0.473972 0.880540i \(-0.342820\pi\)
0.473972 + 0.880540i \(0.342820\pi\)
\(642\) 0 0
\(643\) −13.0278 −0.513765 −0.256882 0.966443i \(-0.582695\pi\)
−0.256882 + 0.966443i \(0.582695\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −23.7889 −0.935238 −0.467619 0.883930i \(-0.654888\pi\)
−0.467619 + 0.883930i \(0.654888\pi\)
\(648\) 0 0
\(649\) −6.42221 −0.252094
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −23.2389 −0.909407 −0.454703 0.890643i \(-0.650255\pi\)
−0.454703 + 0.890643i \(0.650255\pi\)
\(654\) 0 0
\(655\) 6.00000 0.234439
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −13.3944 −0.521774 −0.260887 0.965369i \(-0.584015\pi\)
−0.260887 + 0.965369i \(0.584015\pi\)
\(660\) 0 0
\(661\) 34.8444 1.35529 0.677645 0.735389i \(-0.263001\pi\)
0.677645 + 0.735389i \(0.263001\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 9.39445 0.364301
\(666\) 0 0
\(667\) 4.18335 0.161980
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 19.3944 0.748714
\(672\) 0 0
\(673\) 25.0278 0.964749 0.482375 0.875965i \(-0.339774\pi\)
0.482375 + 0.875965i \(0.339774\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 21.6333 0.831436 0.415718 0.909494i \(-0.363530\pi\)
0.415718 + 0.909494i \(0.363530\pi\)
\(678\) 0 0
\(679\) −20.8444 −0.799935
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −9.00000 −0.344375 −0.172188 0.985064i \(-0.555084\pi\)
−0.172188 + 0.985064i \(0.555084\pi\)
\(684\) 0 0
\(685\) −16.8167 −0.642531
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −10.6056 −0.404039
\(690\) 0 0
\(691\) −9.60555 −0.365412 −0.182706 0.983168i \(-0.558486\pi\)
−0.182706 + 0.983168i \(0.558486\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 4.00000 0.151729
\(696\) 0 0
\(697\) −7.39445 −0.280085
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 11.5778 0.437287 0.218644 0.975805i \(-0.429837\pi\)
0.218644 + 0.975805i \(0.429837\pi\)
\(702\) 0 0
\(703\) 7.21110 0.271972
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −31.2666 −1.17590
\(708\) 0 0
\(709\) 22.8444 0.857940 0.428970 0.903319i \(-0.358877\pi\)
0.428970 + 0.903319i \(0.358877\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −16.8167 −0.629789
\(714\) 0 0
\(715\) −30.4222 −1.13773
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −37.2666 −1.38981 −0.694905 0.719101i \(-0.744554\pi\)
−0.694905 + 0.719101i \(0.744554\pi\)
\(720\) 0 0
\(721\) 10.4222 0.388143
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.39445 0.0517885
\(726\) 0 0
\(727\) 35.6333 1.32157 0.660783 0.750577i \(-0.270224\pi\)
0.660783 + 0.750577i \(0.270224\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0.972244 0.0359597
\(732\) 0 0
\(733\) −32.0000 −1.18195 −0.590973 0.806691i \(-0.701256\pi\)
−0.590973 + 0.806691i \(0.701256\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.63331 0.133835
\(738\) 0 0
\(739\) 6.02776 0.221735 0.110867 0.993835i \(-0.464637\pi\)
0.110867 + 0.993835i \(0.464637\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 5.57779 0.204629 0.102315 0.994752i \(-0.467375\pi\)
0.102315 + 0.994752i \(0.467375\pi\)
\(744\) 0 0
\(745\) 23.0278 0.843672
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −30.0278 −1.09573 −0.547864 0.836567i \(-0.684559\pi\)
−0.547864 + 0.836567i \(0.684559\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 14.4222 0.524878
\(756\) 0 0
\(757\) −16.7889 −0.610203 −0.305101 0.952320i \(-0.598690\pi\)
−0.305101 + 0.952320i \(0.598690\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −11.4500 −0.415061 −0.207530 0.978229i \(-0.566543\pi\)
−0.207530 + 0.978229i \(0.566543\pi\)
\(762\) 0 0
\(763\) −18.2389 −0.660291
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 9.21110 0.332594
\(768\) 0 0
\(769\) 41.0000 1.47850 0.739249 0.673432i \(-0.235181\pi\)
0.739249 + 0.673432i \(0.235181\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −4.81665 −0.173243 −0.0866215 0.996241i \(-0.527607\pi\)
−0.0866215 + 0.996241i \(0.527607\pi\)
\(774\) 0 0
\(775\) −5.60555 −0.201357
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −16.6056 −0.594956
\(780\) 0 0
\(781\) −34.0555 −1.21860
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 17.8167 0.635904
\(786\) 0 0
\(787\) 39.0278 1.39119 0.695595 0.718434i \(-0.255141\pi\)
0.695595 + 0.718434i \(0.255141\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −39.6333 −1.40920
\(792\) 0 0
\(793\) −27.8167 −0.987798
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 41.6611 1.47571 0.737855 0.674959i \(-0.235839\pi\)
0.737855 + 0.674959i \(0.235839\pi\)
\(798\) 0 0
\(799\) −14.7889 −0.523194
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 58.0555 2.04873
\(804\) 0 0
\(805\) −7.81665 −0.275501
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −43.3944 −1.52567 −0.762834 0.646595i \(-0.776193\pi\)
−0.762834 + 0.646595i \(0.776193\pi\)
\(810\) 0 0
\(811\) −13.5778 −0.476781 −0.238390 0.971169i \(-0.576620\pi\)
−0.238390 + 0.971169i \(0.576620\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −2.00000 −0.0700569
\(816\) 0 0
\(817\) 2.18335 0.0763856
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 0 0
\(823\) −17.8167 −0.621050 −0.310525 0.950565i \(-0.600505\pi\)
−0.310525 + 0.950565i \(0.600505\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −48.2111 −1.67646 −0.838232 0.545314i \(-0.816411\pi\)
−0.838232 + 0.545314i \(0.816411\pi\)
\(828\) 0 0
\(829\) 12.7889 0.444177 0.222088 0.975027i \(-0.428713\pi\)
0.222088 + 0.975027i \(0.428713\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0.338936 0.0117434
\(834\) 0 0
\(835\) 3.00000 0.103819
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −33.2111 −1.14657 −0.573287 0.819354i \(-0.694332\pi\)
−0.573287 + 0.819354i \(0.694332\pi\)
\(840\) 0 0
\(841\) −27.0555 −0.932949
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 30.6333 1.05382
\(846\) 0 0
\(847\) −26.6056 −0.914178
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −6.00000 −0.205677
\(852\) 0 0
\(853\) −29.2111 −1.00017 −0.500085 0.865977i \(-0.666698\pi\)
−0.500085 + 0.865977i \(0.666698\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −5.23886 −0.178956 −0.0894780 0.995989i \(-0.528520\pi\)
−0.0894780 + 0.995989i \(0.528520\pi\)
\(858\) 0 0
\(859\) 30.0278 1.02453 0.512267 0.858826i \(-0.328806\pi\)
0.512267 + 0.858826i \(0.328806\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −30.2111 −1.02840 −0.514199 0.857671i \(-0.671911\pi\)
−0.514199 + 0.857671i \(0.671911\pi\)
\(864\) 0 0
\(865\) −10.8167 −0.367777
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −53.4500 −1.81317
\(870\) 0 0
\(871\) −5.21110 −0.176571
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −2.60555 −0.0880837
\(876\) 0 0
\(877\) −22.6611 −0.765210 −0.382605 0.923912i \(-0.624973\pi\)
−0.382605 + 0.923912i \(0.624973\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 25.3944 0.855561 0.427780 0.903883i \(-0.359296\pi\)
0.427780 + 0.903883i \(0.359296\pi\)
\(882\) 0 0
\(883\) −15.8167 −0.532273 −0.266136 0.963935i \(-0.585747\pi\)
−0.266136 + 0.963935i \(0.585747\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 30.6333 1.02857 0.514283 0.857621i \(-0.328058\pi\)
0.514283 + 0.857621i \(0.328058\pi\)
\(888\) 0 0
\(889\) 50.0555 1.67881
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −33.2111 −1.11137
\(894\) 0 0
\(895\) −21.2111 −0.709009
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −7.81665 −0.260700
\(900\) 0 0
\(901\) −2.57779 −0.0858788
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 7.00000 0.232688
\(906\) 0 0
\(907\) −1.57779 −0.0523898 −0.0261949 0.999657i \(-0.508339\pi\)
−0.0261949 + 0.999657i \(0.508339\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 5.02776 0.166577 0.0832885 0.996525i \(-0.473458\pi\)
0.0832885 + 0.996525i \(0.473458\pi\)
\(912\) 0 0
\(913\) 13.8167 0.457265
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −15.6333 −0.516257
\(918\) 0 0
\(919\) 26.4222 0.871588 0.435794 0.900046i \(-0.356468\pi\)
0.435794 + 0.900046i \(0.356468\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 48.8444 1.60773
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −19.3944 −0.636311 −0.318156 0.948039i \(-0.603063\pi\)
−0.318156 + 0.948039i \(0.603063\pi\)
\(930\) 0 0
\(931\) 0.761141 0.0249454
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −7.39445 −0.241824
\(936\) 0 0
\(937\) 41.2111 1.34631 0.673154 0.739502i \(-0.264939\pi\)
0.673154 + 0.739502i \(0.264939\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −0.422205 −0.0137635 −0.00688175 0.999976i \(-0.502191\pi\)
−0.00688175 + 0.999976i \(0.502191\pi\)
\(942\) 0 0
\(943\) 13.8167 0.449932
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 39.0000 1.26733 0.633665 0.773608i \(-0.281550\pi\)
0.633665 + 0.773608i \(0.281550\pi\)
\(948\) 0 0
\(949\) −83.2666 −2.70295
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 30.8444 0.999148 0.499574 0.866271i \(-0.333490\pi\)
0.499574 + 0.866271i \(0.333490\pi\)
\(954\) 0 0
\(955\) 12.4222 0.401973
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 43.8167 1.41491
\(960\) 0 0
\(961\) 0.422205 0.0136195
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0.183346 0.00590212
\(966\) 0 0
\(967\) 50.0000 1.60789 0.803946 0.594703i \(-0.202730\pi\)
0.803946 + 0.594703i \(0.202730\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −18.0000 −0.577647 −0.288824 0.957382i \(-0.593264\pi\)
−0.288824 + 0.957382i \(0.593264\pi\)
\(972\) 0 0
\(973\) −10.4222 −0.334121
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 15.2111 0.486646 0.243323 0.969945i \(-0.421762\pi\)
0.243323 + 0.969945i \(0.421762\pi\)
\(978\) 0 0
\(979\) 63.6333 2.03373
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 42.6333 1.35979 0.679896 0.733309i \(-0.262025\pi\)
0.679896 + 0.733309i \(0.262025\pi\)
\(984\) 0 0
\(985\) −22.8167 −0.726999
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1.81665 −0.0577662
\(990\) 0 0
\(991\) −29.1833 −0.927040 −0.463520 0.886087i \(-0.653414\pi\)
−0.463520 + 0.886087i \(0.653414\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −1.21110 −0.0383945
\(996\) 0 0
\(997\) 39.8722 1.26276 0.631382 0.775472i \(-0.282488\pi\)
0.631382 + 0.775472i \(0.282488\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8640.2.a.df.1.1 2
3.2 odd 2 8640.2.a.cr.1.1 2
4.3 odd 2 8640.2.a.cy.1.2 2
8.3 odd 2 2160.2.a.y.1.2 2
8.5 even 2 135.2.a.d.1.2 yes 2
12.11 even 2 8640.2.a.ck.1.2 2
24.5 odd 2 135.2.a.c.1.1 2
24.11 even 2 2160.2.a.ba.1.2 2
40.13 odd 4 675.2.b.h.649.1 4
40.29 even 2 675.2.a.k.1.1 2
40.37 odd 4 675.2.b.h.649.4 4
56.13 odd 2 6615.2.a.v.1.2 2
72.5 odd 6 405.2.e.k.136.2 4
72.13 even 6 405.2.e.j.136.1 4
72.29 odd 6 405.2.e.k.271.2 4
72.61 even 6 405.2.e.j.271.1 4
120.29 odd 2 675.2.a.p.1.2 2
120.53 even 4 675.2.b.i.649.4 4
120.77 even 4 675.2.b.i.649.1 4
168.125 even 2 6615.2.a.p.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.2.a.c.1.1 2 24.5 odd 2
135.2.a.d.1.2 yes 2 8.5 even 2
405.2.e.j.136.1 4 72.13 even 6
405.2.e.j.271.1 4 72.61 even 6
405.2.e.k.136.2 4 72.5 odd 6
405.2.e.k.271.2 4 72.29 odd 6
675.2.a.k.1.1 2 40.29 even 2
675.2.a.p.1.2 2 120.29 odd 2
675.2.b.h.649.1 4 40.13 odd 4
675.2.b.h.649.4 4 40.37 odd 4
675.2.b.i.649.1 4 120.77 even 4
675.2.b.i.649.4 4 120.53 even 4
2160.2.a.y.1.2 2 8.3 odd 2
2160.2.a.ba.1.2 2 24.11 even 2
6615.2.a.p.1.1 2 168.125 even 2
6615.2.a.v.1.2 2 56.13 odd 2
8640.2.a.ck.1.2 2 12.11 even 2
8640.2.a.cr.1.1 2 3.2 odd 2
8640.2.a.cy.1.2 2 4.3 odd 2
8640.2.a.df.1.1 2 1.1 even 1 trivial